How Should Financial Intermediation Services be

Taxed?¤

Ben Lockwoody

First version: May 31 2010This version: April 17 2012

Abstract

This paper considers the optimal taxation of savings intermediation and payment

services in a dynamic general equilibrium setting, when the government can also use

consumption and income taxes. When payment services are used in strict proportion

to …nal consumption, and the cost of intermediation services is …xed and the same

across …rms, the optimal taxes are generally indeterminate. But, when …rms di¤er

exogenously in the cost of intermediation services, the tax on savings intermediation

should be zero. Also, when household time and payment services are substitutes

in transactions, the optimal tax rate on payment services is determined by the

returns to scale in the conditional demand for payment services, and is generally

di¤erent to the optimal rate on consumption goods. The extension to the case of

commodities is studied, and conditions su¢cient for uniform taxation of goods and

payment services within a period are obtained, generalizing the analysis of Auerbach

and Gordon(2002).

JEL Classi…cation: G21, H21, H25

Keywords: …nancial intermediation services, tax design, banks, monitoring,payment

services

¤I would like to thank Steve Bond, Clemens Fuest, Michael Devereux, Michael McMahon, Miltos

Makris, Ruud de Mooij, and seminar participants at the University of Southampton, GREQAM, the

2010 CBT Summer Symposium, and the 2011 IIPF Conference for helpful comments on an earlier draft.

I also gratefully ackowledge support from the ESRC grant RES-060-25-0033, "Business, Tax and Welfare".yCBT, CEPR and Department of Economics, University of Warwick, Coventry CV4 7AL, England;

Email: B.Lockwood@warwick.ac.uk

1. Introduction

Financial intermediation services include such important services as intermediation be-

tween borrowers and lenders, insurance, and payment services (e.g. credit and debit card

services). These services comprise a signi…cant and growing part of the national economy;

for example, …nancial intermediation services, measured using the OECD methodology1,

were 3.9% of GDP in the UK in 1970, and increased to 7.9% by 2005. The …gures for the

Eurozone countries as a whole are 2.7% to 5.5%. In the US, the …nance and insurance

sector, excluding real estate, which includes …nancial intermediation, accounted for 7.3%

of US value-added in 1999, rising to 8.4% in 20092.

The question of whether, and how, …nancial intermediation services should be taxed

is a contentious one. In the tax policy literature, it is largely assumed that within a

consumption tax system, such as a VAT, it is desirable to tax …nancial services. For

example, the European Commission has recently proposed changes to the VAT treatment

of …nancial services within the European Union, so as bring these more within the scope of

VAT (de la Feria and Lockwood (2010)). Also, the recent IMF proposals for a "bank tax"

to cover the cost of government interventions in the banking system include a Financial

Activities Tax levied on bank pro…ts and remuneration, which would work very much like

a VAT, levied using the addition method (IMF(2010)).

But, it is also recognized that there are technical di¢culties in taxing …nancial inter-

mediation when those services are not explicitly priced (so-called margin-based services),

such as the intermediation between borrowers and lenders. This raises a problem for the

use of a VAT via the usual invoice-credit method, for example (Ebril, Keen, Bodin and

Summers(2001)). As a result of this, the status quo in most countries is that a wide range

of …nancial intermediation services are not taxed3. However, conceptually, the problems

can be solved, for example, by use of a cash-‡ow VAT (Ho¤man et. al.(1987), Poddar and

English(1997), Huizinga(2002), Zee(2005)), and the increasing sophistication of banks’ IT

systems means that these solutions are also becoming practical.

So, it is increasingly relevant to ask, setting aside the technical problems, should

…nancial intermediation services supplied to households be taxed at all? And if so, at

1See http://www.euklems.net.2See http://www.bea.gov/industry/gdpbyind_data.htm3For example, in the EU, the Sixth VAT Directive and subsequent legislation exempts a wide range

of …nancial services from VAT, including insurance and reinsurance transactions, the granting and the

negotiation of credit, transactions concerning deposit and current accounts, payments, transfers, debts,

cheques, currency, bank notes and coins used as legal tender etc. (Council Directive 2006/112/EC of 28

November 2006, Article 135).

2

what rates? Given the overall importance of …nancial services to modern economies,

there is surprisingly little written on this more fundamental question (Section 2 has a

discussion of the literature). Moreover, we would argue that the existing literature does

not really clarify which of the fundamental principles of tax design apply. For example, is

it the case that …nancial intermediation services are intermediate goods in the production

of …nal consumption for households, and thus should not be taxed? Or, should they be

taxed at the same rate as other goods purchased on the market, at least under conditions

when a uniform consumption tax is optimal?

The objective of this paper is to address these fundamental questions4. We set up and

solve the tax design problem in a dynamic general equilibrium model of the Chamley(1986)

type, where the government chooses taxes on payment services and savings intermediation,

as well as the usual taxes on consumption (or equivalently, wage income) and income from

capital, and where …nancial intermediaries, in the form of banks, are explicitly modelled.

On the payment services side, we assume, following the literature on the transactions cost

approach to the demand for money, that payment services are not necessarily proportional

to consumption, but can be used to economize on the household time input to trading.

This is realistic: for example, making use of a basic bank account requires a time input,

e.g. trips to the bank, but use of an additional payment service e.g. a credit card,

substitutes for trips to the bank.

We assume that the cost of savings intermediation per unit of capital is …xed, but can

vary across borrowers (…rms). Again, this is realistic; savings intermediation is a complex

process involving initial assessment of the borrower via e.g. credit scoring, structuring and

pricing the loan, and monitoring compliance with loan covenants (Gup and Kolari(2005,

chapter 9). The extension to variable costs of savings intermediation is addressed in

Lockwood(2010).

We then solve the tax design problem, where the government has access to a full set of

taxes, i.e. the usual wage and capital income taxes, plus a tax on the consumption good

and on payment services, and a tax paid by the bank on the spread between borrowing and

lending rates. We set the wage tax equal to zero to eliminate the usual tax indeterminacy

via the household budget constraint, and focus on the four remaining taxes.

The tax on savings intermediation is determined as follows. In the tax design problem,

the tax on capital income is used as the instrument to pin down the rate of substitution

4It should be noted that this paper does not deal with corrective taxes on bank lending designed to

internalize the social costs of bank failure or the costs of bailout; on this, see e.g. Hellmann, Murdock

and Stiglitz(2000), Keen(2010) or Bianchi and Mendoza(2010).

3

between present and future consumption for the household. So, this means that the

tax on savings intermediation is a "free instrument" that can be used to ensure that

capital is allocated e¢ciently across …rms. In turn, the cost of capital to a particular

…rm will be the cost of capital to the bank i.e. the return paid to depositors, plus the

cost of intermediation, where the latter includes any tax. A non-zero tax on savings

intermediation will distort the relative cost of capital across …rms, and so this tax is

optimally set to zero. This is a version of the Diamond-Mirrlees production e¢ciency

result.

Turning to the tax on the payment service, our …rst result is that the total tax "wedge"

between consumption and leisure is a weighted average of the tax on consumption and

on the payment service, and is determined by a standard optimal tax formula, involv-

ing the general equilibrium expenditure elasticity of consumption (Atkeson, Chari, and

Kehoe(1999)). However, the sign of the tax on the payment service itself is determined5

not by the structure of preferences, but by the properties of the conditional demand for

payment services as a function of the household consumption level and time input to

transactions, or "shopping time". In particular, when this conditional demand has con-

stant returns with respect to these variables, payment services should be untaxed; this

can be understood as an instance of the Diamond-Mirrlees production e¢ciency result.

The general conclusion is that the tax on payment services is determined in a completely

di¤erent way to the tax on consumption, and thus will in general be at a di¤erent rate.

We then extend the analysis to many consumption goods in each period. We make the

plausible assumption that there is only one kind of payment service that must be taxed

at the same rate, whichever good it is used to purchase. We show that the presence of

payment services generally makes it less likely that commodity taxation will be uniform

within the period. Moreover, even if conditions hold for commodity taxation to be uni-

form, this does not imply that payment services need to be taxed at the same uniform

rate as commodities. Generally, what is required for the same uniform tax on both goods

and payment services are: (i) standard conditions for uniformity i.e. separability in goods

and leisure, plus a homothetic goods sub-utility function, plus either (ii) no time input

to transactions, or (iii) that time and payment services are not substitutable in the pro-

duction of tranactions services i.e. Leontief technology, and the ratio of payment services

to time required is uniform across goods. These results generalize, and demonstrate the

limitations of, the Auerbach-Gordon(2002) result that if goods are subject to a uniform

5Strictly speaking, this requires the conditional demand for payment services to be Cobb-Douglas, but

it is also likely to hold for a variety of other cases, see Section 4.

4

tax, payment services should be subject to the same tax.

The remainder of the paper is organized as follows. Section 2 discusses related lit-

erature. Section 3 outlines the model, and explains how existing contributions can be

viewed as special cases. Section 4 presents the main results. Section 5 studies the case of

commodities, Section 6 considers other extensions, and Section 7 concludes.

2. Related Literature

There is a small literature directly addressing the optimal taxation of borrower-lender

intermediation and payment services, Grubert and Mackie(1999), Jack(1999), Auerbach

and Gordon (2002), and Boadway and Keen(2003). Using for the most part a simple two-

period consumption-savings model, these papers broadly agree on a policy prescription6.

Given a consumption tax that is uniform across goods (at a point in time), payment

services should be taxed at this uniform rate, but savings intermediation should be left

untaxed. The argument used to establish this is simple; in a two-period consumption-

savings model with the same, exogenously …xed, tax on consumption in both periods,

this arrangement leaves the marginal rate of substitution between current and future

consumption undistorted i.e. equal to the marginal rate of transformation7.

However, one can make three criticisms of the current literature. First, even taking

their set-up as given, their optimal taxes are indeterminate. Purely mathematically,

two taxes cannot be uniquely determined from a single e¢ciency condition. Second, in

their analysis, consumption (wage) and capital income taxes are taken as given, and not

optimized by the government. Third, relative to the model of this paper, the models

analyzed in the current literature are very special in a number of respects. For example,

implicitly, these papers are assuming8 a …xed labour supply, so that a uniform tax on

consumption over the life-cycle is …rst-best e¢cient, as it does not distort the inter-

temporal allocation of consumption, and thus …nancial intermediation should not do so

6Chia and Whalley(1999), using a computational approach, reach the rather di¤erent conclusion that

no intermediation services should be taxed, but but their model is not directly comparable to these others,

as the intermediation costs are assemed to be proportional to the price of the goods being transacted.7Auerbach and Gordon(2002) have a model that is in some respects more general, and they also take

a di¤erent analytical approach. Speci…cally, their model allows for periods, multiple consumption

goods, and variable labour supply. In this setting, they show that a uniform tax on all commodities

and payment services is equivalent to a wage tax. Thus, they show that if a uniform commodity tax is

optimal, payment services should be taxed at the same rate, consistently with the other literature cited.8The exception here is Auerbach and Gordon(2002), where labour supply is variable. However, in

their model, the consumption tax is just assumed to be uniform, not optimised.

5

either. Again, special assumptions are made about the demand for payment services, and

intermediation activities of banks. Speci…cally, they assume (i) that payment services

are consumed in proportion to consumption; (ii) that the costs of savings intermediation

are in proportion to capital invested. We are able to show that the basic result of this

literature - i.e. that intermediation taxes are indeterminate, but that an optimal tax

structure is to tax payment services at the same rate as consumption, but exempt savings

intermediation - also emerges in our model when all of these special assumptions are

made (Proposition 1 below).

A less closely related literature is that on the optimal in‡ation tax which takes a

transaction costs approach to the demand for money (Kimbrough(1986), Guidotti and

Vegh(1993), Correia and Teles(1996, 1999)). In this literature, money formally plays

a role similar to payment services in our model; the main di¤erences are (i) that it is

assumed a free good i.e. it has a zero production cost, and (ii) it is subject to an in‡ation

tax, rather than a …scal tax. While (ii) makes no di¤erence from an analytical point of

view, (i) does; it turns out that when money is free, the optimal in‡ation tax is zero, as

long as the transactions demand for household time is a homogenous function of money

and consumption. A much more closely related …nding is in Correia and Teles (1996),

where, in Section 3 of their paper, money is allowed to have a positive production cost.

Proposition 3 below can be regarded as an extension of Proposition 2 in their paper.

Finally, one can interpret the use of payment services and time as inputs to household

production of …nal consumption goods, so our analysis is linked to the small literature on

optimal taxation with household production, particularly Kleven(2004). These links are

explored further in Section 6.1 below.

3. The Model

3.1. Households

The model is a version of Atkeson, Chari and Kehoe(1999) with payment services and

savings intermediation. There is a single in…nitely lived household with preferences over

levels of a single consumption good, leisure, and a public good in each period = 0 1 of

the form1X

=0

(( ) + ()) (3.1)

where is the level of …nal consumption in period is the consumption of leisure,

and is public good provision. Utilities ( ) () are strictly increasing and strictly

6

concave in their arguments.

We take a transactions cost approach to the demand for payment services9, and sup-

pose that consumption incurs a transaction cost in terms of household time, and this

cost is reduced by the use of payment services For example, making use of a bank

account requires a time input, e.g. trips to the bank, but use of an additional payment

service e.g. a credit card substitutes for trips to the bank. Then we have = ( )

where is increasing in and decreasing in

To help intuition, consider a special case where consumption level requires sep-

arate transactions, and where a fraction of these transactions are undertaken from a

transactional account with a credit card, and the remainder,1 ¡ from a transactional

account without a credit card, where visits to the bank to withdraw cash are required.

Also assume for simplicity that there is no interest penalty with a transactional account

i.e. it pays the same as a savings account10. Then, the amount of payment services re-

quired is = + where is the …xed cost of maintaining a credit card account,

and is the amount of payment services needed per transaction. Also, the household time

needed is =(1¡)

where is the number of transactions …nanced by a single trip to

the bank, assumed …xed. Eliminating from between these two equations gives a linear

time demand function

( ) =1

µ ¡

+

¶

(3.2)

It turns out that for our purposes, it is convenient to describe the implicit relationship

= ( ) between and in terms of the conditional demand for payment

services

= ( ) 0 · 0 (3.3)

In the case of the linear time demand function, (3.2), takes the form

( ) = + ¡ (3.4)

Speci…cation (3.4) is interesting not only because it has plausible microfoundations, but

also because it nests the existing literature as a special case: this literature e¤ectively

assumes independent of and linear homogenous in i.e. = This is of course

9This is of course, analagous to the transactions cost theory of the demand for money.10This is, to a …rst approximation, a reasonable assumption; transactions accounts do pay similar rates

of interest to instant access savings accounts. It is also analytically very convenient: without it, a version

of the Baumol-Tobin inventory demand theory would come into play, implying that conditional on

would also depend on which are determined in general equilibrium, and this would greatly

complicate the analysis.

7

a special case of (3.4) where there are no …xed costs ( = 0) = and no account

without a credit card. In the general case, following Guidotti and Vegh(1993), we will

assume that is convex in its arguments; this ensures that the household problem is

concave.

The household thus supplies labour to the market of amount

= 1¡ ¡ (3.5)

where the total endowment of time per period is set at unity. In each period the

household also saves +1 in units of the consumption good, and deposits it with a bank,

who can then lend it on to …rms who can use it as an input to production in the next

period, after which they must repay the loan to the bank, who then in turn repays the

household. So, in this model, capital only lasts one period. Finally, the household has no

pro…t income in any period: …rms may generate pure pro…ts (see Section 3.2 below), but

these are taxed at 100%.

In any period household is assumed to pay ad valorem taxes on and

also pays proportional taxes on labour and capital income. Using the well-known fact

that an uniform consumption tax (i.e. = ) is equivalent to a wage tax, we assume

w.l.o.g. that the wage tax is zero We also assume for convenience that one unit of the

consumption good can be transformed into one unit of payment services or one unit of

the public good. Moreover, in equilibrium, payment services are priced at marginal cost

(see Section 3.3 below). This …xes the relative pre-tax price of and at unity.

So, the present value budget constraint of the household is

1X

=0

((1 + ) + +1 + (1 + )) =1X

=0

((1¡ ) + (1 + )) (3.6)

where is the price of output in period is the after-tax return on capital to the

household, and is the wage. We normalize by setting 0 = 1 and assume for convenience

that 0 = 0 i.e. initial capital is zero11 Finally, = (1 ¡ ) where is the pre-tax

return on capital, determined below, and is the capital income tax.

Substituting (3.3),(3.5) in (3.6) gives:

1X

=0

((1 + ) + +1 + ( )(1 + )) =1X

=0

((1¡ ¡ ) + (1 + )) (3.7)

11This simpli…es the implementability constraint, and does not change anything of substance (see

Atkeson, Chari and Kehoe(1999).

8

The …rst-order conditions for a maximum of (3.1) subject to (3.7) with respect to +1

respectively are:

= (1 + + (1 + )) (3.8)

= (3.9)

¡(1 + ) = (3.10)

= (1 + +1)+1 (3.11)

where is the multiplier on (3.7), and we use (here and below) the notation that for

any any function and variables , the partial derivative of with respect to

is the cross-derivative is etc. Note that using this notation, the consumer price

of …nal consumption is (1 + + (1 + )) a weighted sum of the prices facing the

household of and

3.2. Firms

There are …rms, = 1 with produce the homogenous good in each period Firm

produces output from labour and capital via the strictly concave, decreasing returns,

production function () where

are capital and labour inputs. We assume

that …rms face decreasing returns, because for …xed policy, they will generally face di¤erent

costs of capital, but the same wage, and so in this environment, with constant returns,

only the one …rm with the lowest unit cost would operate. This case is of limited interest

because then, a spread tax cannot a¤ect the relative cost of capital of di¤erent …rms,

which means in turn that there is no e¢ciency argument for setting the spread tax to

zero.

These …rms are assumed to be perfectly competitive. But, they cannot purchase capital

directly from households, but must borrow from banks. Moreover, we suppose that …rms

may di¤er in intermediation costs, as described in more detail in Section 3.3 below. So,

…rms face di¤erences in the cost of capital i.e. …rm must repay 1+ per unit of capital

borrowed from the bank. Thus, pro…t-maximization implies:

(

) =

(

) = 1 + (3.12)

And, in addition, the capital and labour market clearing conditions are:X

=1

= X

=1

= 1¡ ¡ (3.13)

These conditions (3.12),(3.13) jointly determine and , given household savings and

labour supply decisions.

9

3.3. Banks

Banks in this economy provide two possible services. First, they can provide payment

services to the households i.e. supply Second, they can provide intermediation be-

tween households and …rms. Banks can compete on price for both these activities (i.e.

households see the banks as perfect substitutes, both with respect to payment and in-

termediation services). We also assume no economies of scope, and constant returns in

the provision of both services, so that banks must break even on both services. Assum-

ing w.l.o.g. that the marginal and average cost of payment services is 1 in units of the

consumption good, the price of payment services will also be 1 in equilibrium.

The cost of intermediating one unit of savings between the household and …rm is .

Note that we take as …xed, but possibly varying between …rms, for reasons discussed in

the introduction. We also suppose that "spread" i.e. the value of intermediation services

provided by the bank, can be taxed at some rate ~ In turn, the value of intermediation

services is measured by ¡ where is the lending rate to …rm and is the rate

paid to depositors. So, is a tax on both intermediation services provided to households,

and to …rms12. Then, as banks make zero pro…t on this activity, we must have

(1¡ )( ¡ )¡ = 0 = 1 (3.14)

Then, from (3.14):

= + (1 + ) =

1¡

(3.15)

We refer to as the spread tax from now on.

3.4. Discussion

The above model provides a general framework which encompasses the speci…c models of

taxation of …nancial services (Auerbach and Gordon(2002), Boadway and Keen(2003)),

Jack(1999), Grubert and Mackie(1999)) that have been developed so far. For example,

Boadway and Keen(2003)), Jack(1999), Grubert and Mackie(1999) are two-period versions

of the above model13, with (implicitly) …xed labour supply. Auerbach and Gordon(2002)

12In principle, one could allow for the intermediation services received by these two parties to be

taxed at di¤erent rates, but in practice, this is very di¢cult to implement (Poddar and English(1997)).

Moreover, in our framework, the addition of this feature in our model would lead to tax indeterminacy.13A minor quali…cation here is that Boadway and Keen allow for a …xed cost of savings intermediation

e.g. …xed costs of opening a savings account. These introduce a non-convexity into household decision-

making, which greatly complicates the optimal tax problem, and so we abstract from these in this paper.

10

is a …nite-horizon version of the model, with the additional feature14 that there are

consumption goods in each period. This raises some new issues, and the case of goods

is covered in Section 5 below.

As already noted in Section 2, the feature of all these contributions, however, is the

special assumptions they implicitly make about demand for payment services and bank

intermediation. On the household side, they all assume, …rst, that payment services are

needed in …xed proportion to consumption and that (implicitly) that a time input is

not required from the household. In our model, this amounts to the assumptions that

( ) = in (3.3), in which case, choosing the constant to be unity, = On

banking activity, the existing literature assumes that the cost of intermediation in …xed

proportion to household savings. In the context of our model, this requires = i.e.

…rms are all the same with respect to intermediation costs, or - equivalently - there is

only one …rm.

Finally, the relation of our model to the optimal in‡ation tax literature is as fol-

lows. Our modelling of household demand for intermediation services is closely related

to the "transactions cost" view of the demand for money in that literature (Corriea and

Teles(1996), (1999)). In particular, if we de…ne as real money balances, their trans-

actions cost function is an inversion of (3.3) to obtain as a function of ; then,

increased real money balances reduce the labour transactions costs of consumption. The

models in this literature do not allow for physical capital or taxation of capital income,

or costly money, and so in this sense are more special. Nevertheless, one of our results,

Proposition 3 below, is related to that literature, especially Proposition 2 of Correia and

Teles(1996).

3.5. A Benchmark Indeterminacy Result

Here, we make the assumptions of the existing literature (Auerbach and Gordon(2002),

Boadway and Keen(2003)), Jack(1999), Grubert and Mackie(1999)), namely: (i) that

conditional demand for is independent of and linear in i.e. = ; (ii) only one

type of …rm; and (iii) a …xed consumption tax and a zero capital income tax = 0.

Under these assumptions, we show that optimal taxes on …nancial intermediation are

generally indeterminate. Note from (3.8)-(3.11) that given (i) i.e. = 1 and = 0 we

have:¡1

=1 + +(1 + )

1 + ¡1 +(1 + ¡1)

1

1 + = 1 (3.16)

14It also has labour supply in only one period.

11

Moreover, from (3.12), (3.15), given only one …rm:

= ¡ 1¡ (1 + ) = 1 (3.17)

where = ( ) Then (3.16) becomes

¡1

=1 + +(1 + )

1 + ¡1 +(1 + ¡1)

1

¡ (1 + ) = 1 (3.18)

Now say that the sequence f g1=0 is a restricted optimal tax structure on …nancial

services if the inter-temporal allocation of consumption is left undistorted by taxes. From

(3.18), this requires:

1 + +(1 + )

1 + ¡1 +(1 + ¡1)

1

¡ (1 + )=

1

¡ = 1 (3.19)

Then two conclusions that can easily be drawn from (3.19). First, f g1=0 is not

uniquely determined from (3.19) i.e. there is indeterminacy in the restricted optimal tax

structure. The second is that of the many optimal tax combinations, = = 0

has the advantage that it is optimal, independently of knowledge of and is thus

administratively convenient. We can thus summarize:

Proposition 1. In the benchmark case, with (i) conditional demand for independent

of and linear in ; (ii) only one type of …rm; and (iii) a …xed consumption tax and

zero capital income tax = 0 then the restricted optimal tax structure on …nancial

services is not uniquely determined. But, a uniform tax on goods and payment services

( = ) and a zero tax on the spread ( = 0) is an administratively convenient

restricted optimal tax structure.

This result summarizes the …ndings of the existing literature, in the context of our

model. It is important to emphasize that under the assumptions made by the existing

literature, optimal taxes on …nancial intermediation are in fact indeterminate. This main

purpose of this paper is to relax these assumptions in an empirically plausible way, and

at the same time generate determinacy in the tax structure.

4. Tax Design

We take a primal approach to the tax design problem. In this approach, an optimal

policy for the government is a choice of all the primal variables in the model, in this case© +1 (

)=1

ª1=0

to maximize utility (3.1) subject to the capital and labour

12

market clearing conditions (3.13), aggregate resource, and implementability constraints.

We are thus assuming, following Chamley(1986), that the government can pre-commit to

policy at = 0 The aggregate resource constraint says that total production must equal

to the sum of the uses to which that production is put:

+ ( ) + +1 + +X

=1

=X

=1

() = 0 1 (4.1)

The implementability constraint ensures that the government’s choices also solve the

household optimization problem. First, by de…nition,

´ + + = ¡ ¡ (4.2)

where is the overhead cost of payment services. The most plausible case is where

there is a …xed cost to payment services i.e. 0 Substituting (4.2) back into (3.7),

we obtain:

1X

=0

((1 + + (1 + )) + (1 + ) + +1) (4.3)

=1X

=0

((1¡ ¡ )¡ (1 + ) + (1 + ))

Then, using the household’s …rst-order conditions (3.8)-(3.11) in (4.3), we …nally arrive

at the government’s implementability constraint:

1X

=0

( ¡ (1¡ + )) = 0 (4.4)

where in (4.4), the expression:

´

(4.5)

is the overhead cost of payment services, normalized by 0 and can thus be inter-

preted as the virtual pro…t of the household from transacting on the market. In the most

plausible case where there is a …xed cost to payment services i.e. 0, virtual pro…t

is negative.

So, as is standard in the primal approach to tax design, we can incorporate the im-

plementability constraint (4.4) into the government’s maximand by writing

= ( ) + () + ( ¡ (1¡ + )) (4.6)

13

where is the Lagrange multiplier on (4.4). If · 0 it is possible to show that ¸ 0

at the optimum (see Appendix). If = 0 the revenue from pro…t taxation is su¢cient

to fund the public good, We will rule out this uninteresting case, and so will assume

that 0 at the optimum in what follows

The government’s choice of primal variables must maximizeP1

=0 subject to

(4.1) and (4.4). The …rst-order conditions with respect to are,

respectively;

= (1 + ) (4.7)

= (4.8)

¡ = + (4.9)

= ¡1 (4.10)

= (4.11)

= +

= 1 (4.12)

= = 1 (4.13)

where are the multipliers on the resource, capital market, and labour market

conditions at time respectively.

Moreover, from (4.6),

= (1 + (1 +)) (4.14)

= ¡ (1¡ + )

and

= (1 + (1 +)) (4.15)

= ¡ (1¡ + )¡

So, is what Atkeson, Chari and Kehoe(1999) call the general equilibrium expenditure

elasticity. Note that if there are constant returns to scale, virtual pro…t ´ 0 and so

are reduce to standard formulae found, for example, in the primal approach to

the static tax design problem (Atkinson and Stiglitz(1980)).

We begin by characterizing the tax on capital income and the spread tax, where we

have a sharp result with strong intuition. It is possible to manipulate the …rst-order

conditions to the household and government optimization problems to get (all proofs in

Appendix):

14

Proposition 2. At any date the optimal taxes satisfy

µ

1 + (1¡ )

µ¡1

¡ 1¡

¶¶

=

¡1

¡1

= 1 (4.16)

where =(1+)(1+

+(1+

))

1+(1+) So, if …rms are homogenous in intermediation costs,

( = all ) then are not uniquely determined, but if there is heterogeneity

( 6= some ) then the unique solution to the system (4.16) has = 0 and in the

steady state, = 0

So, we see that as long as intermediation costs di¤er across …rms, the spread tax at any date should be zero. The intuition for this result is clear. From (4.12), (4.10), we

see that at any date

( ¡ ) =

¡1

=) ¡ =

¡ (4.17)

That is, the marginal product of capital net of true intermediation costs should be equal

across …rms, which of course is just the condition for capital to be allocated e¢ciently

across …rms. But, condition (4.17) is generally not consistent with a non-zero spread tax

when …rms are heterogenous, as then from (3.12), (3.15),

= 1 + + (1 + )

=) ¡ = 1 + +

So, if 6=

(417) cannot hold. This is just an instance of the Diamond-Mirrlees

production e¢ciency theorem. A tax on the spread is an intermediate tax on the allocation

of capital, and given our assumptions (a full set of tax instruments, and no pure pro…ts),

this tax should be set to zero. Note also that when there is only one …rm, this argument

has no bite, and thus is left indeterminate, as in Proposition 1.

Finally, we see that in the steady state, = 0 So, the celebrated result of Cham-

ley(1986) that in the steady state, the tax on capital income is zero continues to hold

in our setting. In this sense, the optimal structure of wage and capital income taxes is

separable from the optimal tax on borrower-lender intermediation.

Next, we turn to characterize the tax on payment services. The …rst step is to charac-

terize the total tax on …nal consumption, which from (3.8) is the weighted sum of and

i.e. + We can then state:

Proposition 3. At any date the optimal total tax on …nal consumption in ad valorem

form is +

1 + + +

=

µ ¡

¶µ ¡

1 +

¶

=

(4.18)

15

Note that (4.18) is a formula for an optimal consumption tax that also occurs in the

static optimal tax problem, when the primal approach is used (Atkinson and Stiglitz(1980,

p377). In particular, is the marginal bene…t of $1 to the government, and is a

measure of the marginal utility of $1 to the household, so ¡

is a measure of the

social gain from additional taxation at the margin. But, inspection of (4.14) and (4.15)

reveals that in our analysis, the are generally di¤erent to the static case, unless

= 0 which occurs when there are constant returns in the conditional demand for

payment services, = 1 Note also that the optimal tax + on …nal consumption is

a weighted average of two taxes on marketed goods, and and thus these two separate

taxes are not yet determinate.

The next result characterizes and can be stated as follows15:

Proposition 4. If household demand for payment services depends on the time input

( 0) any date the optimal ad valorem tax on payment services is

1 +

= ¡

(4.19)

where is the marginal e¤ect of on virtual pro…t (4.5). But, if conditional demand

for payment services is independent of the time input ( = 0) then the optimal tax on

payment services is indeterminate.

That is, generally, is determinate, but under the special conditions of the existing

literature, when = 0 it is not. In the main case of interest, when is determinate, we

see that it is not general equal to the right-hand side of (4.18) but is instead determined

by the e¤ect of on the the virtual pro…t of the household, This of course, implies

that in general, …nancial services should not be taxed at the same rate as the consumption

good i.e. 6= contrary to the claims of the existing literature.

So, how is determined? First, the sign of is the sign of ¡ One intuition

for this is as follows. If the government imposes a positive tax on this will cause a

reduction in and at a …xed level of consumption, a compensating increase in

If this decreases virtual pro…t for the household, which is not directly taxable, this is

desirable. But this last e¤ect is measured just by ¡ Note that in the special case of

constant returns of , then ´ 0 = 0 This can be understood as an instance of the

15As noted in Section 2, Proposition 3 is related to Proposition 2 of Corriea and Teles(1996). They

consider what is formally a very similar tax design problem. The main di¤erences are; (i) Proposition

3 extends their analysis by providing an explicit formula for the optimal tax rate; (ii) they work with a

di¤erent speci…cation of (3.3), namely where is the dependent variable.

16

Diamond-Mirrlees Theorem; if household "pro…t" is zero, the intermediate good, payment

services, should not be taxed.

More generally, there is an analogy here with the Corlett-Hague rule, which says that

goods complementary with non-taxable leisure should be taxed more heavily. An analogy

can also be drawn with tax design when there are non-constant returns to scale in the

production of marketed goods. In that case, it has long been known that in this situation,

a deviation from aggregate production e¢ciency (non-taxation of intermediate goods) is

optimal. For example, Stiglitz and Dasgupta(1971) show that factors of production should

be taxed more heavily when used in industries where pure rent is positive and cannot be

taxed at 100%. Here, the principle is similar: the factor of production, should be

taxed (subsidized), if it causes - indirectly, via - pro…t to rise (fall).

We can now focus on the determinants of the sign of We start with the speci…ca-

tion (3.4), a¢ne conditional demand. In this case, it is clear that =

and thus = 0

implying a zero tax on payment services16. An alternative form for can be derived from

the literature on optimal in‡ation tax, where it is often assumed that ( ) is a ho-

mogenous function (Kimbrough(1986), Guidotti and Vegh(1993), Correia and Teles(1996,

1999)). In this case, without loss of generality17, we can write ( ) = ¡

¢ with

(0) = 0 0 0 This is homogeneous of degree and is convex i¤ 00 0 so we assume

these properties. It is then possible to show:

Proposition 5. Assume ( ) = ¡

¢

(a) If in addition, ¡

¢=

¡

¢is constant elasticity, then at any date the optimal

ad valorem tax on payment services is

1 +

=

(1¡ )

(4.20)

So, if there are decreasing returns to scale ( 1), then is positive, and if there are

increasing returns to scale ( 1), then is negative.

(b) In the general case, at any date the optimal ad valorem tax on payment services

is

1 + =

µ2 +

¡ 1

¶

(1¡ ) = 0

1 =

00

0 0 (4.21)

So, if 2 + and if there are decreasing (increasing) returns to scale, then is

positive (negative).

So, in a wide variety of cases, 0 if there are decreasing returns to scale. This

is the plausible case, as it corresponds to some …xed-cost element of payment services.

16Note that this not a special case of constant returns, as (3.4) is not constant returns in 17This is true because ( 1) = ¡( ) so ( ) = ( 1) ´ ( )

17

But, we cannot rule out a negative tax on payment services. So, overall, the conclusion is

that while the optimal tax on payment services is likely to be positive or zero, a subsidy

cannot be ruled out.

5. Many Consumption Goods

So far, we have assumed only one …nal consumption good for the household. With many

consumption goods, the new issue is when it is optimal to have uniform commodity

taxation within the period and what this implies for taxation of payment services. We

now extend the baseline model to accommodate many consumption goods. To focus on

payment services, we assume that there is only one …rm, for whom the cost of savings

intermediation, is zero. In each period, the consumer has preferences over goods,

(c ) c = (1 )

To keep things reasonably simple, we suppose that demand for payment services

used for purchase of good takes the Cobb-Douglas form

= ( ) = +

¡

¸ 0 0 = 1 (5.1)

where is the amount of time services that are used, and is the returns to scale

in the conditional demand for . This encompasses the case studied by Auerbach and

Gordon(2002), who (implicitly) assume that = 0 and = 118

If every consumption good has its own dedicated payment service, then Propositions

3 and 4 carry over basically unchanged. However, this is very unrealistic; in practice,

households use just a small number of checking accounts, credit cards, etc. So, we focus

on more realistic scenario where there is just type of one payment service, which must

be taxed at the same rate in all its uses i.e. every must be taxed at the rate As

we will see, this imposes additional constraints on the tax design problem. The time and

present-value budget constraints are modi…ed in the obvious way to

= 1¡ ¡X

=1

(5.2)

and1X

=0

(X

=0

(1 + ) +X

=0

( )(1 + )) + +1) =1X

=0

( + (1 + )) (5.3)

18See page 412 of their paper, where it is stated that "time costs..do not enter into the budget constraint

as written here..", implying = 0 and where there is a …xed transaction cost per unit of the good

consumed, requiring that transactions cost are linear in

18

respectively. In (5.3), the government now has separate commodity taxes for each of

the goods.

Then, the household chooses f( )=1 +1g

1

=1 to maximizeP1

=0 (c )

subject to (5.1), (5.2), (5.3). It is then straightforward to set up the government’s tax

design problem, in particular the implementability constraint, much as in the previous

Section. The main di¤erence is that because all the must be taxed at the same rate,

from (A.16), the household’s …rst-order condition for the choice of the derivatives of

( ) with respect to denoted must all be the same across goods, so we need

to impose the additional constraints on the government that

=~ = 1 (5.4)

The government can, however, choose ~ because this is the equivalent, in the

primal problem, of choosing the tax This problem can again be solved using the

primal approach (see Appendix B). Note that the overall e¤ective commodity tax on

good in period is +

where is the derivative of ( ) with respect to

We can prove the following generalizations of Propositions 3 and 4:

Proposition 6. (i) At any date the optimal total ad valorem tax on …nal consumption

good is

+ ¡ ( ¡ )~

1 + + (1 + )=

µ ¡

¶µ ¡

1 +

¶

= (5.5)

where are de…ned in (29) (30) in Appendix B, and are the multipliers on

(5.4), and =1

P=1

(ii) At any date the optimal ad valorem tax on payment services is

1 +

=

1

X

=1

(1¡ )

1

¡1

X

=1

( + 1)( ¡ )~

(5.6)

In the special case where (1¡)

1= = 1 then

1 +

=

(5.7)

So, comparing Proposition 6 with Propositions 3 and 4, we see several qualitative

di¤erences, due to the constraints (5.4). First, the formula for the overall tax on good +

does not precisely follow an Atkinson-Stiglitz type formula, due to the additional

19

e¤ect ¡( ¡ )~ which may be positive or negative, depending on whether the

constraint =~ binds more or less tightly than the average across commodities.

Second, the formula for now depends on the weighted average of economies of scale

in the conditional demand functions, i.e. 1

P=1

(1¡)

1 but unless (1¡)

1= =

1 there is an adjustment given by the last term in (5.6), re‡ecting the additional

constraints (5.4).

We can now address the question of when taxation on …nal consumption goods is

uniform, and what implications this has for the taxation of payment services. There are

two straightforward special cases here. The …rst is where conditional demand (5.1) is

constant returns i.e. = 1 = 1 Then, (1¡)

1= = 0 and so from (5.7),

= 0 all Moreover, from (29) in Appendix B

=1

ÃX

=1

¡ (1¡ )

!

(5.8)

Under certain well-known conditions, this is independent of for example, if (c ) =

(~(c) ) ~ homothetic. It then follows from (5.5) that the tax on the consumption

good, and on …nal consumption overall, is uniform i.e. = at a rate given by the

right-hand side of (5.5)

The second special case is where = so that is independent of in which

case, as in Proposition 2, only the weighted average of and is well-de…ned via (5.5).

In this case, is again de…ned in (5.8), so in this case, the left-hand side of (5.5) is

constant across if (c ) = (~(c) ) ~ homothetic. But then, this uniform …nal

consumption tax can be implemented by a uniform tax on both goods and payment

services, independently of the weights We can summarize as follows:

Proposition 7. Assume (c ) = (~(c) ) ~ homothetic. Then if = 1 all =

= 0 If = so that is independent of all then the ad valorem tax

on …nal consumption, +

1 + + (1 + )

is uniform across goods, and this can be implemented by a common uniform tax across

marketed goods and the payment service i.e. = = all = 1

So, we see that when there is no time input to household production, a uniform

consumption tax on goods and …nancial services at the same rate may be an optimal tax

structure (but not the only one). This directly generalizes the results of Auerbach and

Gordon(2002). Their model, is a …nite horizon version of the model of this section, where

20

additionally, it is assumed that there is no time cost of purchasing any of the goods.

They show that in this environment, if there is a uniform tax already in place on the

marketed goods i.e. = then it is best to set = also19. We have extended this

result by explicitly deriving the optimal tax structure and showing under what conditions

a common uniform tax across marketed goods and the payment service is desirable.

In particular, if there is no time cost of purchasing any of the goods, a uniform tax on

goods and the payment service is optimal under the standard conditions on preferences

required for uniformity i.e. separability in goods and leisure and the goods sub-utility

function homothetic. However, this result does not generally extend to the case where

i.e. time does substitute for payment services. For example, from Proposition 7, we see

that with a Cobb-Douglas speci…cation and constant returns, for example, the optimal

tax on payment services is zero. Thus, the Auerbach and Gordon(2002) result depends

crucially on lack of substitutability between time costs and payment services20.

6. Extensions

6.1. Payment Services as Household Production

Our …ndings in Section 5 also relate to the small literature on optimal taxation with

household production, in particular Kleven(2004), who adopts Becker’s(1965) household

production framework21. First, note that in our framework, the link between the level of

the good actually consumed - in Becker’s notation - and the amount purchased in the

market, along with payment services and time can be written as follows, where we

have dropped time subscripts for simplicity. First, the relationship (3.3), = ( )

can be inverted to give = ( ) Then, the implicit Becker production function for

our model can be written

= min©

( )ª

(6.1)

19In particular, they show that if there is initially a wage income tax at rate which is replaced by a

consumption tax at equivalent rate (1 ¡ ) then the real equilibrium is left unchanged if and only if

transaction services are also taxed at this equivalent rate.20Proposition 8 below implies that uniform taxation of goods and the payment service is also an

optimal tax structure where time is an input to transactions, but where time and payment services are

not substitutable i.e. Leontief technology and the ratio of payment services to time required is uniform

across goods. So, most generally, the Auerbach-Gordon result applies in an environment where time and

payment services are used in …xed proportions, along with other conditions.21Other contributions include Sandmo(1990), Anderberg and Balestrino(2000), but these are less closely

related.

21

That is, actual consumption is the minimum of the actual amount of market good

purchased, and the amount of that can be bought using payment services and

time This implies that the consumer who wishes to consume at level will buy

units of the market good and = ( ) units of payment services, along with time

input

This compares to the version of Becker’s production function studied by Kleven, where

each …nal consumption good is produced via a …xed coe¢cients production function,

where household production of requires a marketed good and time in …xed proportions:

= min

½

¾

(6.2)

and the household cares about …nal goods and leisure i.e. = (1 ) Note the dif-

ference between our structure and Kleven’s. In particular, his can formally be considered

a special case of ours where the only other input to household production besides the

marketed good is time and the production function is linear22. However, in practice,

the interpretations are slightly di¤erent. If good is omelettes, then would be number

of eggs bought in both frameworks, but in Kleven’s framework, would be the time

needed to cook an omelette, whereas in our framework, would be the amount of time

used to buy the eggs.

Nevertheless, due to the formal similarity, it is helpful to compare our optimal tax

rules to Kleven’s. First, under some conditions on preferences, Kleven …nds a very simple

optimal tax structure, where relative taxes across goods just depend on the household

production technology. Speci…cally, given (6.2), the tax on good is just inversely pro-

portional to the share of the marketed good in the total (tax-inclusive) cost of one

unit , where

=(1 + )

(1 + ) +

where is the producer price of and is the ad valorem tax on (Kleven(2004),

Propositions 1-3). In fact, using our primal approach, it is possible to show that in his

framework, the optimal tax is23

1 +

=1

µ ¡

¶( ¡ )

1 +

(6.3)

22Formally, Kleven assumes = minn

o but by approapriate choice of units, we can set = 1

w.l.o.g.23This can be proved along the lines of Proposition 8.

22

where have the standard de…nitions as in (4.14),(4.15). This in fact generalizes

Proposition 3 of Kleven(2004); the latter is the special case of (6.3) where compensated

cross-price e¤ects are zero. Then, if standard conditions for uniform taxes hold i.e. =

(~(1 ) ) where the sub-utility function is homothetic, it is well-known that =

so then is inversely proportional to

To make the link between our results and Kleven’s, assume ( ) = minn

o

in (6.1) i.e. payment services and time must be consumed in strict proportions. Then,

our household production function (6.1) becomes

= min

½

¾

(6.4)

Now, consider the optimal tax problem, as de…ned in Section 5 above, with …xed coe¢-

cients speci…cation (6.4) De…ne

=1 + + (1 + )

1 + + (1 + ) +

to be the share of the (tax-inclusive) cost of producing one unit of that arises from the

purchase of the marketed good and intermediation services. Then, it is straightforward

to show:

Proposition 8. In the case of a …xed coe¢cients production technology, the optimal

overall tax rate on good = ( +

)(1 + ) is given by

1 +

=1

µ ¡

¶( ¡ )

1 +

(6.5)

where are de…ned in (29) (30) in Appendix B with ´ 0 So, if =

(~(1 ) ) with ~ homothetic, then = and so, is inversely proportional

to

Comparing Proposition 8 to (6.3), the formal similarity is clear. Note also that us-

ing Proposition 8, we can state two propositions that relate to Auerbach-Gordon(2002).

First, under the additional assumption that the payment services used in the purchase of

di¤erent commodities can be taxed at di¤erent rates , a possible optimal tax structure

is to always tax good and the payment services used to purchase good at the same

rate i.e. = This is because (6.5) only determines the weighted sum of

Second, under the conditions on preferences in Proposition 8, is independent of

So, assume that at a uniform tax = = = Then, a uniform tax on both

goods and the payment service is optimal. But for = we require that

1 +

=

1 + (6.6)

23

So, most generally, the Auerbach-Gordon result applies in an environment where time

and payment services are used in …xed proportions, and these proportions are the same

across commodities.

6.2. Endogenizing Savings Intermediation Services

We have, so far, treated the service of savings intermediation by banks in rather "black

box" fashion. In particular, we have treated the amount of intermediation services per

unit of capital supplied to …rm , as exogenous. However, it is clear that banks supply

several di¤erent kinds of intermediation services, notably liquidity services (Diamond and

Dybvig(1983)), and monitoring services (Diamond (1991), Besanko and Kanatas(1993),

Holmstrom and Tirole (1997)). If these services provide externalities for the rest of the

economy, then rather than

In this version of the paper, we do not attempt provide a fully microfounded version of

these kinds of intermediation services, for several reasons. First, it is technically di¢cult

to embed some explicit models of intermediation services into the dynamic optimal tax

framework. For example, "endogenize" intermediation by looking at the provision of

liquidity services using Diamond-Dybvig model, which is undoubtedly the pre-eminent

microeconomic model of banking. While this is a topic for future work, the problem

is that the Diamond-Dybvig model has a three-period dynamic structure, which is very

di¢cult to embed within the standard in…nite-horizon dynamic optimal tax model.

Second, the payo¤ from doing so in terms of increased insights is not really proportion-

ate to the increased complexity. In the end, bank intermediation activity, when explicitly

modelled, may (or may not) have spillovers on the rest of the economy. If there are

spillovers, then the optimal tax is a Piguovian one to internalise these spillovers. Ulti-

mately, this is because the government can use the interest income tax to control the

household’s marginal rate of substitution between present and future consumption, and

so any tax on intermediation services is a free instrument which can be used to internalize

externalities arising from bank activity.

These general points are illustrated in a previous version of the paper, Lockwood(2010),

where is interpreted as the level of bank monitoring, along the lines of Holmstrom and

Tirole(1997). In their framework, without monitoring, bank lending to …rms is impossi-

ble, because the informational rent they demand is so high that the residual return to the

bank does not cover the cost of capital. So, as monitoring is costly, the socially e¢cient

level of monitoring is that level which just induces to bank to lend. In the case where

the bank is competitive, i.e. where …rm chooses the terms of the loan contract subject

24

to a break-even constraint for the bank, an assumption commonly made in the …nance

literature, this is also the equilibrium level of monitoring. In this case, savings interme-

diation should not be taxed, because doing to will violate production e¢ciency, as in the

case with heterogenous …rms and a …xed amount of intermediation services per unit of

savings. But, in the case where the bank is a monopolist i.e. it chooses the contract, it

will generally choose a higher level of monitoring than this, in order to reduce the …rm’s

informational rent. So, in this case, the optimal tax is a positive Pigouvian tax, set to

internalize this negative externality.

7. Conclusions

This paper has considered the optimal taxation of two types of …nancial intermediation

services (savings intermediation, and payment services) in a dynamic economy, when the

government can also use wage and capital income taxes. When payment services are used

in strict proportion to …nal consumption, and the cost of intermediation services is the

same across …rms, the optimal taxes on …nancial intermediation are generally indetermi-

nate. But, when …rms di¤er in the cost of intermediation services, the tax on savings

intermediation should be zero. Also, when household time and payment services are sub-

stitutes in household "production" of …nal consumption, the optimal tax rate on payment

services is determinate, and is generally di¤erent to the optimal rate on consumption

goods.

We then extended the analysis to many consumption goods in each period. We show

that the presence of payment services generally makes it less likely that commodity taxa-

tion will be uniform within the period. Moreover, even if conditions hold for commodity

taxation to be uniform, this does not imply that payment services need to be taxed at

the same uniform rate as commodities. These results generalize, and demonstrate the

limitations of, the Auerbach-Gordon(2002) result that if goods are subject to a uniform

tax, payment services should be subject to the same tax.

There are two obvious limitations of the analysis. The …rst is that the government

is assumed to be able to precommit to a tax policy at time zero. However, even in

a simpler setting without a banking sector, the characterization of the optimal time-

consistent capital and labour taxes is a technically demanding exercise (see e.g. Phelan

and Stacchetti (2001)) and so such an extension is certainly beyond the scope of this

paper.

The second is the restriction to linear income taxation. The classic result of Atkin-

son and Stiglitz tells us that with non-linear income taxation, commodity taxation is

25

redundant, and more recently, Golosov et. al. (2003) has recently shown that this result

generalizes to a dynamic economy. Their result would apply, for example, in a version of

our model where households di¤er in skill levels, and without any …nancial intermedia-

tion. What would happen if we introduced …nancial intermediation in this environment?

The results on taxation of payment services seem likely to be a¤ected, as the government

has additional degrees of freedom with which to tax the notional "pro…t" from household

production.

26

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29

A. Appendix

A.1. Proofs of Propositions

Proof that ¸ 0 Suppose to the contrary that 0 at the optimum. Then, from the

properties of ( ) 0 from (4.14). So, from (4.14), But from (4.8),(4.11),

(4.13), (3.12):

= = =

So, combining the two, we see that

(A.1)

But, (A.1) says that utility could be increased if 1$ of spending on the public good were

returned to the household as a lump-sum, contradicting the optimality of the policy. ¤

Proof of Proposition 2. From (4.7), (4.15), we get

¡1

=1 + 1 + ¡1

¡1¡1

=¡1

¡1

(A.2)

where =1+

1+(1+) Next, from from (4.10),(4.12),

¡ =

=

¡1

(A.3)

So, combining (A.2) and (A.3), we get

¡1

=

¡1(

¡ ) (A.4)

Next, using (3.8), (3.11), = (1¡ ) and (3.15), we get:

¡1

=¡1

(1 + (1¡ )) =

¡1

¡1 + (1¡ )

¡ ¡ 1¡ (1 + )

¢¢

(A.5)

where = 1 + + (1 + ) Combining (A.4), A.5), and eliminating +1

we get

that:¡1 + (1¡ )

¡ ¡ 1¡ (1 + )

¢¢=

¡1(

¡ ) = 1 (A.6)

where = Finally, using (A.3) to substitute ¡ by

¡1

in (A.6), we get

(4.16) as required. If = 1 (4.16) is a single condition and thus are not uniquely

determined. If 1 (4.16) comprises a system of 1 equations, and it is easy to

30

verify that = 1¡

¡1

¡1

¡1

¡1

¡1 = 0 is the unique solution to this system. So, = 0

is a solution in the steady state. ¤

Proof of Proposition 3. From (4.7), (4.8), (4.13),(4.14),(4.15),(3.12), we have

=

1 + (1 +)

1 + (1 +)=1 +

(A.7)

And, from (3.8),(3.9):=1 + + (1 + )

(A.8)

So, combining (A.7), (A.8) we get:

+

1 + + (1 + )=

( ¡)

1 + (1 +)(A.9)

Also, from (4.8),(4.11),(4.13),(4.14),(3.12) we have:

(1 + (1 +)) = = (A.10)

=) =1

1 +

¡

=

Combining (A.9),(A.10) to eliminate and rearranging, we get (4.18) as required. ¤

Proof of Proposition 4. From (3.10), we have

1 +

= +

(A.11)

And from (4.9), (4.13), (3.12), we get:

¡

= +

= +

(A.12)

But then, combining (A.11),(A.12) and using (4.11) and =

we get

¡

=

1 +

as required. ¤

Proof of Proposition 5. Drop the "t" subscripts for convenience. As is homogenous

of degree

( ) =(1¡ )

= ¡

¡

¢

0¡

¢2

(1¡ )

31

Then, di¤erentiating, and cancelling terms,

= 1¡ ¡2

0

(1¡ )¡ 00

( 0)2(1¡ )

So,

¡ = (1¡ )

µ

¡1 +2

+

¶

(A.13)

Substituting (A.13) into (4.19), we obtain (4.21), as required. Note that 0 00 0 implies

0 and from convexity of () 0() all so 1 Finally, if ¡

¢=

¡

¢

it is easy to compute that 2+

¡ 1 = 1 so (4.20) then follows from (4.21)¤

A.2. The Many-Good Case

The household chooses f1 1 +1g1=0 to maximize

P1=0

(c ) subject

to:

1X

=0

(X

=1

((1 + ) + ( )(1 + ))+ +1) =1X

=0

(1¡ ¡X

=1

) + (1+ ))

(A.14)

The FOC with respect to +1 respectively are:

= (1 + + (1 + )) = 1 (A.15)

¡(1 + ) = = 1 (A.16)

= (A.17)

= (1 + +1)+1 (A.18)

where denotes the derivative of with respect to Combination of (A.15-A.18)

with (A.14) gives the implementability constraint for government. This can be derived

following the same steps as in the one-good case, by substituting (A.15-A.18) and

´ + +

= ¡ ¡ (A.19)

in (A.14) to give:

1X

=0

Ã

¡(1¡ ) +X

=1

¡ ¡

¢!

= 0

where

=

(A.20)

32

is the overhead cost of payment services, normalized by 0 and can thus be inter-

preted as the virtual pro…t of the household from transacting on the market. In the most

plausible case where there is a …xed cost to payment services i.e. 0, virtual pro…t

is negative.

So, the government choosesn1 1 +1 ~

o1

=0to maximize

1X

=0

Ã

(1 ) + (1 ) +

ÃX

=1

¡ ¡

¢¡ (1¡ )

!!

subject to the resource constraints

( + ( )) + + +1 = (1¡ ¡X

=1

) (A.21)

and the constraints

=~ = 1 (A.22)

The FOC for this problem are :

= (A.23)

= (1 + ) + = 1 (A.24)

¡ +

= ( + ) = 1 (A.25)

¡1 = (A.26)

= (A.27)X

=1

= 0 (A.28)

where are the multipliers on (A.44), (A.22) respectively.

Proof of Proposition 6. (i) De…ne and implicitly by

=X

=1

¡ ¡

Ã

1¡ +X

=1

!

(A.29)

=X

=1

¡ ¡

¢¡ (1¡ ) (A.30)

where etc. denote cross-partials. Then from (A.23), (A.24), (3.12) we have:

=

1 + (1 +)

1 + (1 +)=1 + + ~

(A.31)

33

And, from (A.15),(A.17):=1 + + (1 + )

(A.32)

So, combining (A.49), (A.50), we get:

+ ¡ ~

1 + + (1 + )=

( ¡ )

1 + (1 +)(A.33)

Then, using (A.10) to substitute for in (A.51) which still applies in this case, and using

(A.28), we get (5.5) as required.

(ii) From (A.16), (A.22), and = we have

1 +

= + ~

(A.34)

and from (A.25), (A.27), and = , we get:

¡

+

=

1 +

= 1 (A.35)

Moreover, from (A.20), we get

=( ¡ 1)

µ

()2

¡ 1

¶

(A.36)

We can compute, using (A.22), that

()2=1

+ 1 = ¡( + 1)

~

(A.37)

Substituting (A.36),(A.37) into (A.35), we get

1 +

=

(1¡ )

1

¡( + 1)( ¡ )~

Averaging this across all goods, and using = 0 from (A.28), we get (5.6) as required.

Finally, in the case where (1¡)

1= holds, we can argue as follows. Consider the

less-constrained problem for the government where we do not impose (5.4) and so the can vary across commodities. In that case, it is easy to check, using the argument above,

that

1 + =

(1¡ )

1

= 1 (A.38)

34

So, if (1¡)

1= holds, from (A.38),

1 +

=

= 1 (A.39)

in the less-constrained problem, and so (A.39) must also solve the original problem. ¤

Proof of Proposition 8. From (6.4), the consumer will choose = = .

So, the household problem is to choose f1 +1g1=0 to maximize

P1=0

(c )

subject to:

1X

=0

(X

=1

((1 + ) + (1 + ) +) + +1) =

1X

=0

(1¡ ) + (1 + )) (A.40)

The FOC with respect to +1 respectively are:

= = (1 + ) + (1 + ) + ) = 1 (A.41)

= (A.42)

= (1 + +1)+1 (A.43)

Combination of (A.41-A.43) with (A.14) gives the implementability constraint for gov-

ernment. This can be derived similarly to the proof of Proposition 5 to give:

1X

=0

ÃX

=1

¡ (1¡ )

!

= 0

So, the government chooses f1 +1 g1=0 to maximize

1X

=0

Ã

(1 ) + (1 ) +

ÃX

=1

¡ (1¡ )

!!

subject to the resource constraints

(1 + ) + + +1 = (1¡ ¡X

=1

) (A.44)

The FOC for this problem are :

= (A.45)

= (1 + + ) = 1 (A.46)

¡1 = (A.47)

= (A.48)

35

where is the multipliers on (A.44) as before.

(b) Now, de…ne and by setting ´ 0 in (A.29) ,(A.30). Then from (A.45),

(A.46), (3.12) we have:

=

1 + (1 +)

1 + (1 +)=(1 + + )

(A.49)

And, from (A.15),(A.17):

=1 + + (1 + ) +

(A.50)

So, combining (A.49), (A.50), we get:

+

1 + + (1 + ) + =

( ¡)

1 + (1 +)(A.51)

Then, using (A.10) to substitute for in (A.51) which still applies in this case, we get

+

1 + + (1 + ) + =

µ ¡

¶( ¡ )

1 +

But then

+

1 + + +

=1 + + + +

1 + + +

µ ¡

¶( ¡ )

1 +

=)

1 + =1

µ ¡

¶( ¡ )

1 +

as required. ¤

36